# Universal Property of Ext sheaves

This is a problem from Hartshorne's Algebraic Geometry Chapter III Ex.6.4:

Let $$X$$ be a noetherian scheme, and suppose that every coherent sheaf on $$X$$ is a quotient of a locally free sheaf. Then for any $$\mathcal{G} \in \mathcal{Mod}(X)$$, show that the $$\delta$$-functor $$\mathscr{Ext}^{i}(\cdot,\mathcal{G})$$ from $$\mathcal{Coh}(X)$$ to $$\mathcal{Mod}(X)$$, is a contravariant universal $$\delta$$-functor.

My attempt and failure:

By the hint from this book, we want to show that this $$\delta$$-functor is coeffacable. Now by condition we have a surjection from a locally free sheaf $$\mathcal{F'}$$ to $$\mathcal{F}$$. From Hartshorne's book we know that locally free sheaf of finite rank has vanished higher Ext sheaves, so I want to find a locally free sheaf of finite rank of $$\mathcal{F'}$$. By extension of coherent sheaf we can find a coherent subsheaf of $$\mathcal{F'}$$. But it seems that this sheaf not necessary becomes locally free again, and neither has a vanished higher Ext sheave from a priori.

Any help is appreciated.

Edit: I've thought about that Hartshorne implies the locally free sheaf to have finite rank since he also refers it as "$$\mathcal{Coh}(X)$$ has enough locally frees". If this is the case I appreciate a counter example of the original statement above.